Everything you were taught at university about the weirdness of quantum mechanics, from cats that are simultaneously dead and alive to reality waiting politely for an observer to look at it before deciding what to do, hangs on a single word. In 1926 Max Born suggested that the squared modulus of the wave function, |Ψ|², gives the probability of finding a particle at a given place when you make a measurement. In 1927, at the fifth Solvay conference in Brussels, Louis de Broglie pointed out that if you change the word finding to being, every paradox evaporates. Both entities exist at all times: a real wave field Ψ evolving smoothly without collapse according to the Schrödinger equation, and real particles at definite positions guided along continuous trajectories by the gradient of the wave’s phase. David Bohm rediscovered the full many-body theory in 1952, John Bell championed it for thirty years, and most physics departments carried on pretending it did not exist.
Once you take the trajectories seriously, however, a remarkable physical question opens up. Because the de Broglie–Bohm velocity field carries the probability density |Ψ|² along with the wave, an ensemble of systems distributed as |Ψ|² today will remain distributed as |Ψ|² forever, reproducing every statistical prediction of textbook quantum mechanics. In pilot-wave theory, therefore, the Born rule is not a fundamental law of nature written on stone tablets; it is a state of statistical equilibrium, quantum equilibrium, strictly analogous to Maxwell’s thermal distribution in a classical gas. As Antony Valentini showed in 1991 with his subquantum H-theorem, an ensemble started out of equilibrium (P ≠ |Ψ|²) is stirred by the complex phase gradient of the wave until, on any coarse-grained scale, it relaxes towards |Ψ|².
From toy boxes to real matter: why CASINO is here
CASINO’s involvement in this story began in 2010 when I wrote LOUIS, a trajectory code designed to test Valentini’s relaxation theorem for a single particle in a two-dimensional box. In our 2012 paper (Proc. R. Soc. A 468, 990), Nick Russell, Antony Valentini and I showed how the nodes of a multi-mode wave function act as microscopic quantum vortices, winding an out-of-equilibrium probability density into ever finer filaments until it matches the Born rule. LOUIS has shipped as a utility inside CASINO since 2013, and in today’s CASINO v3.2 release it gains a new manual and a full ensemble driver.
Yet for thirty-five years, a glaring gap remained in the foundations literature. Every numerical study of quantum relaxation had been performed in a toy model: a particle in a box, a harmonic oscillator, or a free scalar field. Nobody had ever followed an out-of-equilibrium ensemble in a real, correlated many-electron atom, molecule or crystal. The reason was simple: foundations theorists did not have correlated many-electron wave functions, and the quantum Monte Carlo community, who spend their lives building Slater–Jastrow and multideterminant wave functions within fractions of a millihartree of exact eigenstates, never used them to integrate real-time trajectories. At every step of a variational or diffusion Monte Carlo run, CASINO already computes the exact complex vector ∇i ln Ψ for every electron: its real part is the familiar DMC drift vector, and its imaginary part is the de Broglie–Bohm guidance velocity! We had the machinery sitting in the code all along.
In CASINO v3.2, that machinery is now unleashed. Setting RUNTYPE = pilot_eval turns CASINO into a stateless, multi-state wave-function evaluator that computes ln|Ψ|, the phase, the local energy, the local kinetic energy (which, for a real state away from its nodes, is the Bohmian quantum potential Q), and the logarithmic gradients for almost any wave function CASINO knows how to read. The companion driver pilotdrive performs subspace diagonalisation across a set of correlated VMC trial states and integrates the real-time trajectories of an ensemble using an adaptive fifth-order Dormand–Prince stepper, or, when a diffusion constant is switched on, integrates Edward Nelson’s 1966 real-time stochastic mechanics. Evaluating the wave functions scales gently as about N1.6, taking tens of microseconds per step in helium and a few milliseconds in a 64-electron crystal supercell.
What real electrons taught us: from helium to diamond
I’ve just completed the first study of de Broglie–Bohm and Nelson relaxation across a five-rung ladder of real correlated matter: the helium atom (2 electrons in superpositions of up to ten singlet states within 0.6 mHa of exact), the beryllium atom (4 electrons, 12 states), the water molecule (10 electrons, 6 states), the benzene molecule (42 electrons, 4 states), and a periodic diamond crystal (64 electrons in a 2×2×2 supercell, 5 states). Turning QMC trial wave functions into real-time pilot waves immediately uncovered three pieces of physics that no toy box could show:
- The dimensional cascade. From beryllium to diamond, the initial drop in the coarse-grained H-function accelerates dramatically as the dimension 3N of the configuration space grows. Within just 20 atomic units of time (about half a femtosecond), H̄ falls to a third of its initial value in beryllium, an eighth in water, and under a fiftieth in benzene and diamond, with the 64-electron diamond crystal reaching the statistical sampling floor of a Born ensemble by the second recorded step (5 a.u.).
- Spontaneous Loschmidt echoes on a phase torus. In a few-electron atom like helium, adding more states does not guarantee faster relaxation. Because bound atomic eigenstates are real, whenever the relative phases 2(En − E1)t of the superposed states pass near multiples of 2π on their torus, a condition predictable from the energy spectrum alone before running a single trajectory, the wave function becomes nearly real up to a global phase. At that “near-real instant”, the velocity field becomes odd in time, v(t* + τ) ≈ −v(t* − τ): every electron comes to a halt and retraces its own path backwards through six-dimensional space! In five-state helium, the ensemble relaxes steadily for 556 atomic units down to a tenth of its initial H̄, hits the near-real instant at t* = 555.9 a.u., and dutifully climbs back up its own mirrored decay curve for the next 170 a.u. Under Nelson’s stochastic dynamics, by contrast, the osmotic drift and Brownian noise are not odd under time reversal: the noise obliterates the Loschmidt echo, fills the diffuse Rydberg halo by diffusion, and relaxes water and benzene six to eight times faster than the deterministic flow.
- The continuity defect and the Real-Jastrow Theorem. Because QMC trial wave functions are not exact eigenstates, differentiating |Ψ|² along a Bohmian trajectory reveals a small continuity leak whose rate ε(R, t) is driven by the exact local-energy fluctuations EL(n)(R) − En that variational Monte Carlo minimises. Integrating that defect along each trajectory yields an exact pathwise weight eS that restores the Born distribution. Even better, because a common real Jastrow factor eJ has zero phase, it cancels out of Im ∇ ln Ψ for a single state: the Jastrow factor carves a 1-Hartree correlation hole around each electron in the Bohmian quantum potential Q while leaving a single state’s velocity field completely untouched!
The films: imaginary time, real time, and attosecond charge migration
Having exact trajectories and walker histories on disk also allowed us to build a new rendering pipeline linking CASINO and Blender. When watching these animations, one fundamental distinction should be kept in mind.
The films of diffusion Monte Carlo (DMC), among them those of caffeine, C60 and small molecules whose nuclei are quantum particles, are on the Videos and Simulations page, with interactive 3D browser scenes. In those the motion is a computational random walk in imaginary time τ towards the ground state; in real time, the electrons of a real non-degenerate ground state stand completely still where the repulsive quantum force −∇Q balances the electrostatic force −∇V.
The five films below cross over into real time, showing what electrons actually do when guided by Nelson’s stochastic mechanics and by the deterministic de Broglie–Bohm pilot wave. Turn the sound up: every one of them has sound but the film of beryllium.
Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.
pilotdrive, first in one molecule and then in 296 drawn together, whose electrons sample the ion’s density at the start and at every beat, the approximations letting the sample drift from it between beats; a second of film is about 167 attoseconds. Chlorine’s spin-orbit coupling is left out, and in the real molecule the nuclei move and the beat is expected to fade within a few femtoseconds.Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.
Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.
Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.
Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.
Where to find the code and the details
The kinship between imaginary-time DMC walks and real-time pilot-wave and Nelson trajectories is not merely a curiosity. Analysing the stochastic behaviour of walkers near nodal surfaces has already led in CASINO v3.2 to an exact three-dimensional Bessel-process near-node propagator for DMC. You can explore more films and interactive browser visualisations on the Videos and Simulations page, the background lectures on the de Broglie–Bohm page, the technical specifications of pilot_eval and pilotdrive in Chapter 16 (“The wave function outside QMC”) and Chapter 20 of the new CASINO v3.2 manual, and the commit-by-commit history in the development log.
